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## G = C8⋊2SD16order 128 = 27

### 2nd semidirect product of C8 and SD16 acting via SD16/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8⋊2SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8⋊4Q8 — C8⋊2SD16
 Lower central C1 — C22 — C42 — C8⋊2SD16
 Upper central C1 — C22 — C42 — C8⋊2SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊2SD16

Generators and relations for C82SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a-1, cbc=b3 >

Subgroups: 272 in 92 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×2], D4 [×10], Q8 [×2], C23 [×2], C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×3], D8 [×4], SD16 [×4], C2×D4 [×6], C2×Q8, C4×C8, C8⋊C4, D4⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C4×Q8, C41D4 [×2], C2×D8 [×2], C2×SD16 [×2], C4.D8 [×2], C82C8, C84Q8, C4⋊SD16 [×2], C84D4, C82SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8⋊C22 [×3], C4⋊SD16, C82D4, D4.4D4, C82SD16

Character table of C82SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 16 16 2 2 2 2 4 8 8 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ6 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 1 -1 1 linear of order 2 ρ9 2 2 2 2 0 0 2 -2 2 -2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 -2 2 -2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ14 2 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 -2i 0 0 2i 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 0 -2 0 2 0 0 0 0 0 -2 2 0 √-2 0 -√-2 √-2 0 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 0 -2 0 2 0 0 0 0 0 -2 2 0 -√-2 0 √-2 -√-2 0 √-2 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 -2 0 2 0 0 0 0 0 2 -2 0 √-2 0 √-2 -√-2 0 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 0 0 -2 0 2 0 0 0 0 0 2 -2 0 -√-2 0 -√-2 √-2 0 √-2 complex lifted from SD16 ρ19 4 -4 -4 4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 4 -4 0 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ21 4 -4 4 -4 0 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 4 -4 -4 0 0 0 0 0 0 0 0 0 2√2 0 0 -2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 -2√2 0 0 2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4

Smallest permutation representation of C82SD16
On 64 points
Generators in S64
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 25 39 63 24 45 53 10)(2 28 40 58 17 48 54 13)(3 31 33 61 18 43 55 16)(4 26 34 64 19 46 56 11)(5 29 35 59 20 41 49 14)(6 32 36 62 21 44 50 9)(7 27 37 57 22 47 51 12)(8 30 38 60 23 42 52 15)
(2 8)(3 7)(4 6)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 48)(16 47)(17 23)(18 22)(19 21)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 64)(33 51)(34 50)(35 49)(36 56)(37 55)(38 54)(39 53)(40 52)

G:=sub<Sym(64)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,25,39,63,24,45,53,10)(2,28,40,58,17,48,54,13)(3,31,33,61,18,43,55,16)(4,26,34,64,19,46,56,11)(5,29,35,59,20,41,49,14)(6,32,36,62,21,44,50,9)(7,27,37,57,22,47,51,12)(8,30,38,60,23,42,52,15), (2,8)(3,7)(4,6)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,48)(16,47)(17,23)(18,22)(19,21)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,64)(33,51)(34,50)(35,49)(36,56)(37,55)(38,54)(39,53)(40,52)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,25,39,63,24,45,53,10)(2,28,40,58,17,48,54,13)(3,31,33,61,18,43,55,16)(4,26,34,64,19,46,56,11)(5,29,35,59,20,41,49,14)(6,32,36,62,21,44,50,9)(7,27,37,57,22,47,51,12)(8,30,38,60,23,42,52,15), (2,8)(3,7)(4,6)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,48)(16,47)(17,23)(18,22)(19,21)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,64)(33,51)(34,50)(35,49)(36,56)(37,55)(38,54)(39,53)(40,52) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,25,39,63,24,45,53,10),(2,28,40,58,17,48,54,13),(3,31,33,61,18,43,55,16),(4,26,34,64,19,46,56,11),(5,29,35,59,20,41,49,14),(6,32,36,62,21,44,50,9),(7,27,37,57,22,47,51,12),(8,30,38,60,23,42,52,15)], [(2,8),(3,7),(4,6),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,48),(16,47),(17,23),(18,22),(19,21),(25,63),(26,62),(27,61),(28,60),(29,59),(30,58),(31,57),(32,64),(33,51),(34,50),(35,49),(36,56),(37,55),(38,54),(39,53),(40,52)])

Matrix representation of C82SD16 in GL8(𝔽17)

 16 0 16 0 0 0 0 0 0 16 16 2 0 0 0 0 2 0 1 0 0 0 0 0 1 16 0 1 0 0 0 0 0 0 0 0 5 5 7 7 0 0 0 0 12 12 0 10 0 0 0 0 5 12 0 0 0 0 0 0 12 0 12 0
,
 5 5 0 0 0 0 0 0 12 5 0 0 0 0 0 0 7 7 7 10 0 0 0 0 7 0 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 16 16 1 2 0 0 0 0 1 0 0 0 0 0 0 0 16 0 1 1
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 15 0 16 0 0 0 0 0 16 16 16 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 16 16

G:=sub<GL(8,GF(17))| [16,0,2,1,0,0,0,0,0,16,0,16,0,0,0,0,16,16,1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,5,12,5,12,0,0,0,0,5,12,12,0,0,0,0,0,7,0,0,12,0,0,0,0,7,10,0,0],[5,12,7,7,0,0,0,0,5,5,7,0,0,0,0,0,0,0,7,12,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,2,0,1],[1,0,15,16,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16] >;

C82SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("C8:2SD16");
// GroupNames label

G:=SmallGroup(128,420);
// by ID

G=gap.SmallGroup(128,420);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,387,352,1123,136,2804,718,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations

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